Texture Changes When Object Position Changes
The problem you are having is that the two inputs to the Wood Rings Texture are two world-space points that the "core" of the wood passes through.

You are currently supplying a world-space point (the object location, aka center), and a normal vector. However, the normal vector is being used as a point, not a vector. As a result, you are defining a line which passes through the object-center and a point within 1 unit of the global origin.

One part of the solution is to convert the calculations into object space, making these two core endpoints object space coordinates. This can be done by going into the distance node and replacing Geometry->Position with TextureCoordinate->Object. This supplies the object-space position of the texel to the distance calculation.



Then we need to remove the object position, because the core-definition points are now in object space. We can define core-origin as an arbitrary x,y,z point in object space. Further, the normal node (aka core orientation) needs to be added to the core-origin to create the second core-endpoint. 



Here is a fixed blend file.

This solution is a bit limited, in that it has only one core orientation per shader, and we'd prefer to have one per object instance. 

One alternative is to use script nodes to read the object positions of empties, which can be created per object instance. 

Another alternative is to randomly generate core orientations per object.